# Capacitance of semi-cylindrical plates?

For a visual reference of what I mean by semi-cylindrical plates check the diagram below.

EDIT: The above diagram is not what I intend to implement. It is just to show anyone trying to answer the problem what I mean by semi-cylindrical plates.

How would one go about deriving the formula for such a capacitor? I could start with something but don't know how to proceed further. Here is what I could come up with.

If we start with the equation of parallel plate capacitor $$C = \dfrac{\epsilon A}{d}$$

and replace d = f(x), x being the angle shown in the diagram. Then by the formula of chord length $$f(x) = 2rsin(x)$$

this makes the original equation $$C = \dfrac{\epsilon A}{2rsin(x)}$$

• From Wikipedia "... two thin parallel conductive plates each with an area of A separated by a uniform gap of thickness d. It is assumed the gap d is much smaller than the dimensions of the plates". This is not applicable in the above case. The expression may have to be derived from first principles.
– AJN
Jul 15, 2020 at 15:06
• You will want to increase X alot, so that it measures in the middle more than the edge. So just measure it and calibrate it. Since water C, Dk= 80, moisture content and fertilizer (R=?) it is better to use a current source signal then measure voltage as impedance. Jul 15, 2020 at 15:09
• If x is increased, the capacitance is expected to decrease. Will it be large enough to measure ? OP might as well do away with the cylindrical shape and stick with two parallel plates ?
– AJN
Jul 15, 2020 at 15:13
• @AJN regarding your first comment, this presents a further step I might try. I know that the assumed gap d comes from equation dV = E \dot dS which when integrated over the limits of the gap would evaluate to d. So if I were to re-write the chord length formula as a derivative of f(x) w.r.t to x. Then I could substitute it in the voltage equation and derive the capacitance formula. Would this approach be right? I am wondering what the limits of the integration would be in such a case. Jul 15, 2020 at 16:08
• if I were to re-write the chord length formula as a derivative of f(x). I am not sure if it would be the way to go.
– AJN
Jul 15, 2020 at 16:17